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| Fig. 1: Gaussian distrubution described by Eq. (1). (Image source: E. Daly) |
In the field of photonics, the Gaussian beam is the "gold standard" of laser sources. It represents the lowest-order solution to the paraxial Helmholtz equation and offers the unique advantage of remaining Gaussian at every point along its propagation path. [1] The Gaussian beam is idealized with a bell-shaped intensity profile that tapers off at the ends. (See Fig. 1)
However, a theoretical Gaussian distribution is mathematically defined from negative infinity to positive infinity. In the physical world, optical componentslenses, mirrors, and fiber coresare finite. [2] Consequently, a portion of the beams energy is inevitably "clipped" or lost at the edges. Furthermore, for high-power applications such as laser ablation or lithography, the bell-curve shape of the energy distribution represents a spatial inefficiency, where the low-intensity "wings" of the beam do not contribute to the work threshold. [3]
To quantify the energy loss, we must look at the irradiance distribution. The intensity I(r) of a Gaussian beam as a function of the radial distance r from the propagation axis is given by
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(1) |
where I0 is the peak intensity at the center of the beam and w is the beam radius (or spot size) at which the intensity drops to e-2 (approx. 13.5%) of its peak value. The total optical power (Ptotal) in Ea. (2) contained in the theoretical infinite beam is found by integrating the intensity over the entire transverse plane:
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(2) |
The primary source of loss occurs when the beam passes through an optical aperture of radius a. The power transmitted through this finite aperture, P(a), is the integral of intensity from 0 to a. Solving this integral yields the transmission efficiency equation resulting in [3]
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(3) |
This equation reveals the fundamental conflict of Gaussian optics: To transmit 99% of the energy the aperture radius a must be at least 1.5w (1.5 times the beam radius). To transmit 99.9% of the energy the aperture must be approximately 1.9w. If an optical system is designed with lenses exactly matching the beam size (a = w), the transmission drops to approximately 86.5%. This means 13.5% of the beam's energy is immediately lost to the housing of the lens or the edges of the mirror simply due to the geometry of the distribution.
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| Fig. 2: Top-Hat Distribution. (Image source: E. Daly) |
Beyond simple transmission loss, Gaussian beams are energetically inefficient for applications that require a specific intensity threshold.4 Consider a laser cutting process that requires an intensity of Ith to melt metal. Because the Gaussian beam is peaked. The center may significantly exceed Ith (wasting energy as excess heat) and The "wings" of the beam (where I < Ith) heat the material but fail to cut it. This is distinct from transmission loss; it is application loss. The energy is delivered to the target, but it is effectively useless because it falls below the activation energy of the process. This is why "Top-Hat" or "Flat-Top" beam profiles are often preferred for industrial engineering, despite the difficulty in generating them. The current expanding field for creating these tophat distributions is meta surfaces. Taking the gaussian distribution in Fig. 1 and transforming it inot the tophat distribution in Fig. 2.
Metasurfaces are ultra-thin, two-dimensional optical components composed of dense arrays of sub-wavelength nanostructures, often referred to as "meta-atoms" or pillars. Unlike traditional refractive lenses that rely on macroscopic surface curvature to bend light, metasurfaces manipulate electromagnetic waves by introducing precise, local phase delays at the nanoscopic level. By varying the geometry of these pillars (such as their width or orientation) while keeping their height fixed, engineers can exert complete control over the light's wavefront, polarization, and intensity. This "flat optics" technology allows for the creation of lightweight, compact devices that can be manufactured using standard semiconductor lithography, offering a scalable alternative to bulky glass optics for applications ranging from smartphone sensors to high-efficiency laser beam shaping. [4,5]
Engineers must optimize the ratio a/w. Increasing the aperture size a reduces energy loss, but increases the cost and physical footprint of the system. Conversely, decreasing the beam spot w to fit through a small aperture increases divergence (due to the diffraction limit), causing the beam to expand more rapidly downstream. [3] Therefore, energy loss in Gaussian beams is not merely a defect to be fixed, but a constraint to be managed. We accept the loss of the "wings" as the price for the propagation stability of the "center."
The energy loss associated with Gaussian laser beams is twofold: the diffractive loss at finite apertures and the geometric inefficiency of non-uniform intensity. While the TEM00 mode remains the most convenient for transport through free space, its utilization in power-critical applications often necessitates beam shaping or oversized optics. Understanding the exponential decay of the Gaussian profile allows engineers to calculate exactly how much energy must be sacrificed to maintain the integrity of the optical system.
© Erin Daly. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley-Interscience, 2019).
[2] F. M. Dickey, Ed., Laser Beam Shaping: Theory and Techniques, 2nd Ed. (CRC Press, 2014).
[3] A. E. Siegman, Lasers (University Science Books, 1986).
[4] J. Tang, J. Cheng, and S. Chang, "Terahertz Flat-Top Beam Generation Via Triple-Polarization-Channel Multiplexed Metasurface," Opt. Lasers Eng. 195, 109357 (2025).
[5] R. Yamada et al., "Gaussian to Tophat Beam Shaping Metasurface For Visible Light," IEEE 10289595, Seventeenth Intl. Congress on Artificial Materials for Novel Wave Phenomena, 11 Sep 23.